Paper:

# Adaptive Regulation of Nonlinear Systems by Output Feedback

## Mai Bando and Akira Ichikawa

Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501, Japan

In this paper adaptive regulation by output feedback is considered for a class of single-input/single-output nonlinear systems described by multiple linear models. The adaptive laws are based on the filtered state and input of an adaptive observer. Then a controller is given by a state-dependent Riccati equation, which assures the stability of the adaptive system. Simulation results are given to illustrate the theory.

*J. Robot. Mechatron.*, Vol.20, No.5, pp. 719-725, 2008.

- [1] R. Murray-Smith and T. Johansen, “Multiple Model Approaches to Modelling and Control,” Taylor & Francis London, 1997.
- [2] G. Feng, “A Survey on Analysis and Design of Model-Based Fuzzy Control Systems,” IEEE Transactions on Fuzzy Systems, 14(5), pp. 676-697, 2006.
- [3] H. Katayama and A. Ichikawa, “H
_{∞}control for discrete-time Takagi–Sugeno fuzzy systems,” Int. Journal of Systems Science, 33(14), pp. 1099-1107, 2002. - [4] H. Katayama and A. Ichikawa, “H
_{∞}control for sampled-data nonlinear systems described by Takagi–Sugeno fuzzy systems,” Fuzzy Sets and Systems, 148(3), pp. 431-452, 2004. - [5] M. Nishikawa, H. Katayama, J. Yoneyama, and A. Ichikawa, “Design of output feedback controllers for sampled-data fuzzy systems,” Int. Journal of Systems Science, 31(4), pp. 439-448, 2000.
- [6] K. Tanaka, T. Ikeda, and H. Wang, “Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, H
_{∞}control theory, and linear matrix inequalities,” IEEE Transactions on Fuzzy Systems, 4(1), pp. 1-13, 1996. - [7] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets and Systems, 45(2), pp. 135-156, 1992.
- [8] J. Yoneyama, M. Nishikawa, H. Katayama, and A. Ichikawa, “Output stabilization of Takagi–Sugeno fuzzy systems,” Fuzzy Sets and Systems, 111(2), pp. 253-266, 2000.
- [9] J. Yoneyama, M. Nishikawa, H. Katayama, and A. Ichikawa, “Design of output feedback controllers for Takagi–Sugeno fuzzy systems,” Fuzzy Sets and Systems, 121(1), pp. 127-148, 2001.
- [10] J. Yoneyama, M. Nishikawa, H. Katayama, and A. Ichikawa, “H
_{∞}control for Takagi–Sugeno fuzzy systems,” Int. Journal of Systems Science, 32(7), pp. 915-924, 2001. - [11] M. Bando and H. Nakanishi, “A Stable Approach for Modular Learning and Its Application to Autonomous Aero-Robot,” Journal of Robotics and Mechatronics, 18(1), pp. 44-50, 2006.
- [12] M. Bando and A. Ichikawa, “Adaptive Output Regulation of Nonlinear Systems described by Multiple Linear Models,” Proc. of 9th IFAC Workshop on Adaptation and Learning in Control and Signal Processing 2007, 2007.
- [13] B. Chen, C. Tseng, and H. Uang, “Mixed H 2/H
_{∞}fuzzy output feedback control design for nonlinear dynamic systems: an LMI approach,” Fuzzy Systems, IEEE Transactions on, 8(3), pp. 249-265, 2000. - [14] M. Feng and C. Harris, “Feedback stabilization of fuzzy systems via linear matrix inequalities,” Int. Journal of Systems Science, 32(2), pp. 221-231, 2001.
- [15] H. o. Lam, “Design of fuzzy observer controllers for multivariable uncertain systems using stability and robustness analysis,” Multi. Val. Logic, 5, pp. 391-405, 2000.
- [16] J. Cloutier, C. D’Souza, and C. Mracek, “Nonlinear regulation and nonlinear H-infinity control via the state-dependent Riccati equation technique. I- Theory, II- Examples,” Int. Conf. on Nonlinear Problems in Aviation and Aerospace, 1 st, Daytona Beach, FL, pp. 117-141, 1996.
- [17] E. Erdem and A. Alleyne, “Design of a class of nonlinear controllers via state dependent Riccati equations,” IEEE Transactions on Control Systems Technology, 12(1), pp. 133-137, 2004.
- [18] C. Mracek and J. Cloutier, “Control designs for the nonlinear benchmark problem via the state-dependent Riccati equation method,” Int. Journal of Robust and Nonlinear Control, 8(45), pp. 401-433, 1998.
- [19] W. Langson and A. Alleyne, “A Stability Result with Application to Nonlinear Regulation: Theory and Experiments,” Proc. of the American Control Conf., 5, pp. 3051-3056, 1999.
- [20] J. Curtis and R. Beard, “Ensuring stability of state-dependent Riccati equation controllers via satisficing,” Proc. of the 41st IEEE Conf. on Decision and Control, 3, 2002.
- [21] J. Curtis and R. Beard, “Satisficing: a new approach to constructive nonlinear control,” IEEE Transactions on Automatic Control, 49(7), pp. 1090-1102, 2004.
- [22] K. Narendra and A. Annaswamy, “Stable adaptive systems,” Prentice-Hall, Inc. Upper Saddle River, NJ, USA, 1989.
- [23] G. Tao and P. Ioannou, “Strictly positive real matrices and the Lefschetz-Kalman-Yakubovich lemma,” Automatic Control, IEEE Transactions on, 33(12), pp. 1183-1185, 1988.
- [24] G. Tao and P. Ioannou, “Necessary and Sufficient Conditions for Strictly Positive RealMatrices,” Circuits, Devices and Systems, IEE Proc. G, 137(5), pp. 360-366, 1990.
- [25] Y. Leu, T. Lee, and W. Wang, “Observer-based adaptive fuzzyneural control for unknown nonlineardynamical systems,” Systems, Man and Cybernetics, Part B, IEEE Transactions on, 29(5), pp. 583-591, 1999.
- [26] S. Tong, B. Chen, and Y. Wang, “Fuzzy adaptive output feedback control for MIMO nonlinear systems,” Fuzzy Sets and Systems, 156(2), pp. 285-299, 2005.
- [27] S. Tong, H. Li, and W. Wang, “Observer-based adaptive fuzzy control for SISO nonlinear systems,” Fuzzy Sets and Systems, 148(3), pp. 355-376, 2004.

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